1. Copyright © 2000 by Harcourt, Inc. All rights reserved. Introduction The next two chapters (together with Ch. 2 of Haugen) will briefly examine the following aspects of capital market theory that underlie quantitative investment management: Modeling risk and return – CAPM & APT – theory, testing, and extensions Estimating risk and return – the Single-Index Model (SIM) and multiple-factor models for risk and expected return

2. Lecture Presentation Software to accompany Investment Analysis and Portfolio Management Seventh Edition by Frank K. Reilly & Keith C. Brown Chapters 8 & 9

3. Copyright © 2000 by Harcourt, Inc. All rights reserved. Modeling Risk & Return Part One: The Risk-Free Asset, Portfolio Separation, and The Capital Asset Pricing Model (CAPM)

4. THE RISK-FREE ASSET WHAT IS A RISK FREE-ASSET? – DEFINITION: an asset whose terminal value is certain variance of returns = 0, covariance with other assets = 0 If then

5. THE RISK-FREE ASSET WHAT IS A RISK FREE-ASSET? – DEFINITION: an asset whose terminal value is certain An investment with NO risk An asset with zero variance Zero correlation with all other risky assets Provides the risk-free rate of return (RFR) Will lie on the vertical axis of a portfolio graph

6. THE RISK-FREE ASSET DOES A RISK-FREE ASSET EXIST? – CONDITIONS FOR EXISTENCE: Fixed-income security No possibility of default No interest-rate risk No reinvestment risk

7. THE RISK-FREE ASSET DOES A RISK-FREE ASSET EXIST? – Given the conditions, what qualifies? a U.S. Treasury security with a maturity matching the investor’s horizon

8. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining the Risk-Free Asset with a Risky Portfolio Portfolio expected return is a linear relationship  the weighted average of the two returns

9. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining the Risk-Free Asset with a Risky Portfolio Portfolio standard deviation is also a linear relationship, equal to the weighted average of the two standard deviations (zero for the risk- free asset and  i for the risky portfolio)

10. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining a Risk-Free Asset with a Risky Portfolio Standard deviation The expected variance for a two-asset portfolio is Substituting the risk-free asset for Security 1, and the risky asset for Security 2, this formula would become Since we know that the variance of the risk-free asset is zero and the correlation between the risk-free asset and any risky asset i is zero we can adjust the formula

11. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining a Risk-Free Asset with a Risky Portfolio Given the variance formula the standard deviation is Therefore, the standard deviation of a portfolio that combines the risk-free asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio.

12. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining a Risk-Free Asset with a Risky Portfolio Example Assume: –E(R F ) = 7%, –E(R S&P ) = 12%, –  S&P = 20% Expected Return on Combined Portfolio: Standard Deviation on Combined Portfolio:

13. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining a Risk-Free Asset with a Risky Portfolio Since both the expected return and the standard deviation of return for such a portfolio are linear combinations, a graph of possible portfolio returns and risks looks like a straight line between the two assets. Thus, the existence of a risk-free asset adds value to investors by expanding the set of portfolios available to them.

14. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier Figure 9.1 RFR P* C A B D

15. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Risk-Free Asset and Portfolio Separation Theory Assuming the investor can both lend (by buying Treasury bonds) and borrow (by shorting the bonds with full use of the proceeds) at the risk- free rate, this means that the investor now faces a linear (rather than convex) efficient frontier:

16. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Risk-Free Asset and Portfolio Separation Theory This linear efficient frontier, comprising various combinations of the risk-free asset and the risky portfolio P*, dominates all other possible risky portfolios within the original (Markowitz) efficient frontier. This fact led to the development of the Portfolio Separation Theory (cf., James Tobin).

17. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Separation Theory Under the portfolio separation theory, the ideal risky portfolio in which an investor should invest is the same (P*), regardless of how aggressive or risk averse the investor is. –I.e., the point on the Markowitz efficient frontier at which the investor will invest is independent of the investor’s risk preferences. Where risk preferences are reflected is in terms of how much of his or her portfolio is allocated to P* and how much is invested in the risk-free asset.

18. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Separation Theory Thus, in order to obtain his or her optimal portfolio, there are two separate decisions for the investor to make: 1.The investment decision Which portfolio on the Markowitz efficient frontier to choose? This is determined by the point of tangency between the Markowitz efficient frontier and a line extending from the risk-free rate This leads to the choice of portfolio P* as the optimal risky portfolio for the investor, regardless of the investor’s risk preferences

19. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Separation Theory 2.The financing decision This is where risk preferences come into the picture If the investor is more risk averse, he or she will put part of his or her money in P* and the rest in Treasury bonds (this is known as a lending portfolio, because the rest of the investor’s money is lent to the federal government) If the investor is more aggressive, he or she will leverage up his or her holdings and invest in P* on margin by borrowing at the risk-free rate (this is known as a borrowing portfolio)

20. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier RFR P* Borrowing Lending

21. Copyright © 2000 by Harcourt, Inc. All rights reserved. Capital Market Theory: An Overview Question: What are the general implications for security prices if investors act the way Markowitz portfolio theory and portfolio separation theory say they should? If such theories hold, what would equilibrium in the capital markets entail? Capital market theory extends portfolio theory and develops a model for pricing all risky assets The capital asset pricing model (CAPM) will allow you to determine the required rate of return (for use in discounting future cash flows) for any risky asset

22. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 1. All investors are Markowitz mean-variance efficient investors who want to target points on the efficient frontier. –Also, they include all investable assets in their estimation of the efficient frontier –Not necessarily a realistic assumption! Most investors do not use Markowitz optimization Of those who do, they typically optimize w.r.t. alpha and tracking error rather than mean and variance

23. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 2. Investors can borrow or lend any amount of money at the risk-free rate of return (RFR). –This means that the conditions of portfolio separation theory will hold, at least at the individual level. –Note: it is always possible to lend money at the risk- free rate by buying securities such as T-bills, but (unless you’re the government) it is not usually possible to borrow at this risk-free rate. –However, assuming a higher borrowing rate does not change the general results (although it does change their form a bit).

24. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 3. All investors have homogeneous expectations; that is, investors have identical estimates for the probability distributions of future rates of return. –This implies that all investors will estimate the efficient frontier to be in the exact same location (including using the same risk and expected return factor models), and the optimal portfolio P* (i.e., the investment decision from portfolio separation theory) will be the same for all investors. –This assumption can be relaxed, and as long as the differences in expectations are not vast their effects will be minor.

25. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 4. All investors have the same one-period time horizon such as one-month, six months, or one year. –Markowitz portfolio theory is a single-period model; making the model dynamic requires additional constraints, such as on portfolio turnover, in calculating the efficient frontier. –With regard to capital market theory, differences in investors’ time horizons would require investors to derive risk measures and risk-free assets that are consistent with their time horizons.

26. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 5. Capital markets are “frictionless,” i.e.: –No taxes – true for many classes of investors –No transactions costs – becoming more true over time, but still can be an impediment –Fixed supply of stocks – i.e., don’t have to worry about incorporating IPO shares into the analysis –Infinitely divisible supply of stocks – this assumption allows us to discuss investment alternatives as continuous curves. Changing it would have little impact on the theory, and it is also becoming more true over time. –Information is costless and available to all investors

27. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 6. Capital markets are in equilibrium. –This means that we begin with all investments properly priced in line with their risk levels.

28. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory Note that some of these assumptions are unrealistic, But relaxing many of these assumptions would have only minor influence on the model and would not change its main implications or conclusions; Moreover (as Milton Friedman argues), a theory can be useful for helping to explain and predict behavior, even if not all of its assumptions hold true (e.g., many useful models in physics assume the absence of any friction).

29. Copyright © 2000 by Harcourt, Inc. All rights reserved. Derivation of the Capital Market Line a)Homogeneous expectations (together with the same investment horizon) means that investors all face the same estimated efficient frontier. b)Existence of a risk-free asset means that each investor can mix the riskless asset with a risky portfolio. c)(a) and (b) imply that all investors choose the same risky portfolio to hold in combination with the risk-free asset. Call this portfolio P*

30. Copyright © 2000 by Harcourt, Inc. All rights reserved. Derivation of the Capital Market Line d)In order to have equilibrium (supply = demand), all risky assets must be included in P*. If this were not the case, then some assets would not be held at all. e)In view of (d), the optimal portfolio P* is called the Market Portfolio (M) Value-weighted portfolio, with E(R M ) and  M f)The line connecting RFR with M now represents the market-wide opportunities for expected return and risk. Thus, this line is called the: Capital Market Line (CML)

31. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Capital Market Line (CML) Figure 9.2 RFR M CML Borrowing Lending

32. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Market Portfolio Because portfolio M lies at the point of tangency, it has the highest portfolio possibility line Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML Therefore this portfolio must include ALL RISKY ASSETS (else there will be stocks out there on the market the NO ONE owns!)

33. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Market Portfolio Because the market is in equilibrium, all risky assets are included in this portfolio in proportion to their market value

34. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Market Portfolio Because it contains all risky assets, it is a completely diversified portfolio (once you already own everything, you can’t diversify any more!), which means that all the unique risk of individual assets (unsystematic risk) is diversified away (all the risk that’s left over is, by definition, systematic risk)

35. Copyright © 2000 by Harcourt, Inc. All rights reserved. Systematic Risk Only systematic risk remains in the market portfolio Systematic risk is the variability in all risky assets caused by macroeconomic variables Systematic risk can be measured by the standard deviation of returns of the market portfolio and can (and does) change over time

36. Copyright © 2000 by Harcourt, Inc. All rights reserved. Examples of Macroeconomic Factors that Affect Systematic Risk Variability in growth of money supply Interest rate volatility Variability in: industrial production corporate earnings and cash flow

37. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Market Portfolio and How to Measure Diversification All portfolios on the CML are perfectly positively correlated with each other and with the completely diversified market Portfolio M A completely diversified portfolio would have a correlation with the market portfolio of +1.00 Thus, can use regression R 2 of portfolio’s returns regressed on the “market” portfolio’s returns as a measure of the extent of diversification

38. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Capital Market Line (CML) Describes the risk / return relationship for well- diversified portfolios (idiosyncratic risk has been diversified away). Portfolio standard deviation (  Q ) is the relevant measure of risk, and the portfolio’s expected return (E(R Q )) will be a direct linear function of its risk: To obtain higher expected returns, must accept higher risk.

39. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Security Market Line (SML) Key Question: What is the relevant measure of risk for an individual security when it is held as part of a well diversified portfolio (i.e., the Market portfolio, M)? The Security Market Line describes the risk / return relationship for an individual security. –Also applies to non-diversified portfolios or any other holding for which the total risk may include some diversifiable or idiosyncratic risk.

40. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Security Market Line (SML) The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Cov i,m ) This is the risk measure for the SML, which describes the relationship between risk and expected return for all portfolios, whether well- diversified or not, as well as for all securities The return for the market portfolio should be consistent with its own risk, which is the covariance of the market with itself - or its variance:

41. Copyright © 2000 by Harcourt, Inc. All rights reserved. Graph of Security Market Line (SML) Figure 9.5 RFR SML

42. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Security Market Line (SML) The equation for the risk-return line is We then redefine as beta

43. Copyright © 2000 by Harcourt, Inc. All rights reserved. Graph of SML with Normalized Systematic Risk Figure 9.6 SML Negative Beta RFR

44. Copyright © 2000 by Harcourt, Inc. All rights reserved. Determining the Expected Rate of Return for a Risky Asset The expected rate of return of a risk asset is determined by the RFR plus a risk premium for the individual asset The risk premium is determined by the systematic risk of the asset (  ) and the prevailing market risk premium (R M -RFR) In equilibrium, to obtain higher expected returns, investors must accept higher “covariance” risk In equilibrium, investors receive no compensation for diversifiable (non-systematic or idiosyncratic) risk Q: What is market is not in equilibrium?

45. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Capital Asset Pricing Model: Expected Return and Risk CAPM indicates what should be the expected or required rates of return on risky assets This helps to value an asset by providing an appropriate discount rate to use in dividend (or other discounted cash flow) valuation models Conversely, you can compare an estimated rate of return to the required rate of return implied by CAPM – over / under valued ?

46. Copyright © 2000 by Harcourt, Inc. All rights reserved. Determining the Required Rate of Return for a Risky Asset Assume: RFR = 6% (0.06) R M = 12% (0.12) Implied market risk premium = 6% (0.06) E(R A ) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2% E(R B ) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0% E(R C ) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9% E(R D ) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4% E(R E ) = 0.06 + -0.30 (0.12-0.06) = 0.042 = 4.2%

47. Copyright © 2000 by Harcourt, Inc. All rights reserved. Determining the Required Rate of Return for a Risky Asset In equilibrium, all assets and all portfolios of assets should plot on the SML Any security with an estimated return that plots above the SML is underpriced (or under-valued) Any security with an estimated return that plots below the SML is overpriced (or over-valued) A superior investor must derive value estimates for assets that are consistently superior to the consensus market evaluation to earn better risk-adjusted rates of return than the average investor

48. Copyright © 2000 by Harcourt, Inc. All rights reserved. Identifying Undervalued and Overvalued Assets Compare the required rate of return to the expected rate of return for a specific risky asset using the SML over a specific investment horizon to determine if it is an appropriate investment Independent estimates of return for the securities provide price and dividend outlooks

49. Copyright © 2000 by Harcourt, Inc. All rights reserved. Price, Dividend, and Rate of Return Estimates Table 9.1

50. Copyright © 2000 by Harcourt, Inc. All rights reserved. Comparison of Required Rate of Return to Estimated Rate of Return Table 9.2

51. Copyright © 2000 by Harcourt, Inc. All rights reserved. Plot of Estimated Returns on SML Graph Figure 9.7 SML.20.40.60.801.20 1.40 1.60 1.80 -.40 -.20.22.20.18.16.14.12 R m.10.08.06.04.02 A B C D E

52. Copyright © 2000 by Harcourt, Inc. All rights reserved.